openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /InjectiveCircleReflection.lean
| import OAI.Analysis.CircleDomains.Reflection.SchwarzReflectionInteriorInjectivity | |
| import OAI.Analysis.CircleDomains.Reflection.CayleyHolomorphic | |
| namespace OAI | |
| /-! The circle continuation is injective from interior injectivity alone. | |
| No boundary injectivity or nonzero boundary derivative is assumed. -/ | |
| open Set Function Metric Filter Complex | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_local_injective_circle_reflection {U : Set ℂ} {f : ℂ → ℂ} | |
| (hU : IsOpen U) (hs : ∀z∈U,conj z∈U) | |
| (hc : ContinuousOn f (U∩{z : ℂ | 0≤z.im})) | |
| (hd : ∀z∈U,0<z.im → DifferentiableAt ℂ f z) | |
| (hi : InjOn f (U∩{z : ℂ | 0<z.im})) | |
| (hinside : ∀z∈U,0<z.im → ‖f z‖<1) | |
| (hcircle : ∀z∈U,z.im=0 → ‖f z‖=1) | |
| (hpole : ∀z∈U,0≤z.im → f z≠1) | |
| {a : ℂ} (ha : a∈U) (ha0 : a.im=0) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ a∈V ∧ V⊆U ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ EqOn F f (V∩{z : ℂ | 0≤z.im}) := by | |
| let g : ℂ → ℂ := fun z => diskHalfPlane (f z) | |
| have hgc : ContinuousOn g (U∩{z : ℂ | 0≤z.im}) := | |
| continuousOn_diskHalfPlane.comp hc (fun z hz => hpole z hz.1 hz.2) | |
| have hgd : ∀z∈U,0<z.im → DifferentiableAt ℂ g z := by | |
| intro z hz hz0 | |
| exact (differentiableAt_diskHalfPlane (hpole z hz hz0.le)).comp z (hd z hz hz0) | |
| have hgr : ∀z∈U,z.im=0 → (g z).im=0 := by | |
| intro z hz hz0 | |
| exact diskHalfPlane_real (hpole z hz hz0.ge) (hcircle z hz hz0) | |
| have hgi : InjOn g (U∩{z : ℂ | 0<z.im}) := by | |
| intro z hz w hw he | |
| apply hi hz hw | |
| exact diskHalfPlane_injOn_nonpole (hpole z hz.1 (show 0<z.im from hz.2).le) | |
| (hpole w hw.1 (show 0<w.im from hw.2).le) he | |
| have hgp : ∀z∈U,0<z.im → 0<(g z).im := | |
| fun z hz hz0 => diskHalfPlane_pos (hinside z hz hz0) | |
| let H := schwarzReflect g | |
| have hHi : InjOn H U := injOn_schwarzReflect_of_injOn_upper hU hs hgc hgd hgi hgr hgp | |
| have hHc : ContinuousOn H U := continuousOn_schwarzReflect hs hgc hgr | |
| have hHd : DifferentiableOn ℂ H U := differentiableOn_schwarzReflect hU hs hgc hgd hgr | |
| have hHa : H a+I≠0 := by | |
| change schwarzReflect g a+I≠0 | |
| rw [schwarzReflect_eq ha0.ge] | |
| exact diskHalfPlane_add_I_ne_zero (hpole a ha ha0.ge) | |
| have hnbhd : U∩{z : ℂ | H z+I≠0} ∈ 𝓝 a := by | |
| apply inter_mem (hU.mem_nhds ha) | |
| exact ((hHc.continuousAt (hU.mem_nhds ha)).add_const I).eventually | |
| (isOpen_ne |>.mem_nhds hHa) | |
| obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp hnbhd | |
| refine ⟨ball a ε,fun z => halfPlaneDisk (H z),isOpen_ball,mem_ball_self hε, | |
| fun z hz => (hball hz).1,?_,?_,?_⟩ | |
| · intro z hz | |
| exact ((differentiableAt_halfPlaneDisk (hball hz).2).comp z | |
| ((hHd z (hball hz).1).differentiableAt (hU.mem_nhds (hball hz).1))).differentiableWithinAt | |
| · intro z hz w hw he | |
| apply hHi (hball hz).1 (hball hw).1 | |
| exact halfPlaneDisk_injOn_nonpole (hball hz).2 (hball hw).2 he | |
| · intro z hz | |
| change halfPlaneDisk (schwarzReflect g z)=f z | |
| rw [schwarzReflect_eq hz.2] | |
| exact halfPlaneDisk_diskHalfPlane (hpole z (hball hz.1).1 hz.2) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |